Sequences and Series Quiz

Sequence and Series

Definitions

• Sequence: An ordered list of numbers following a specific pattern.
• Series: The sum of the terms of a sequence.

Types of Sequences

1. Arithmetic Sequence:

   • Difference between consecutive terms is constant.
   • General form: ( a_n = a + (n-1)d )
     • (a): first term
     • (d): common difference
     • (n): term number

2. Geometric Sequence:

   • Ratio between consecutive terms is constant.
   • General form: ( a_n = a \cdot r^{(n-1)} )
     • (r): common ratio

3. Harmonic Sequence:

   • Reciprocals of an arithmetic sequence.
   • General form: ( a_n = \frac{1}{a + (n-1)d} )

4. Fibonacci Sequence:

   • Each term is the sum of the two preceding terms.
   • Starts with 0 and 1: ( F_n = F_{n-1} + F_{n-2} )

Formulas

• nth Term of an Arithmetic Sequence:

  • ( a_n = a + (n-1)d )

• nth Term of a Geometric Sequence:

  • ( a_n = a \cdot r^{(n-1)} )

• Sum of the First n Terms of an Arithmetic Series:

  • ( S_n = \frac{n}{2} (2a + (n-1)d) ) or ( S_n = \frac{n}{2} (a + l) )
    • (l): last term

• Sum of the First n Terms of a Geometric Series:

  • ( S_n = a \frac{1 - r^n}{1 - r} ) (for ( r \neq 1 ))

Key Concepts

• Convergence: A sequence or series converges if it approaches a limit.
• Divergence: A sequence or series diverges if it does not approach a limit.
• Infinite Series: The sum of an infinite sequence, can converge or diverge.

Tests for Convergence

1. Ratio Test: Apply ratios of successive terms.
2. Root Test: Apply root of terms.
3. Integral Test: Compare to integrals.
4. Comparison Test: Compare with known convergent/ divergent series.

Applications

• Used in calculus, physics, finance (e.g., annuities), computer science (e.g., algorithms), and statistics.

Sequences and Series

• Sequences are ordered lists of numbers following a pattern.
• Series are the sums of terms in a sequence.
• Arithmetic Sequences have a constant difference between consecutive terms.
  • The general form is ( a_n = a + (n-1)d ), where (a) is the first term, (d) is the common difference, and (n) is the term number.
• Geometric Sequences have a constant ratio between consecutive terms.
  • The general form is ( a_n = a \cdot r^{(n-1)} ), where (r) is the common ratio.
• Harmonic Sequences are reciprocals of arithmetic sequences.
  • The general form is ( a_n = \frac{1}{a + (n-1)d} ).
• Fibonacci Sequences have each term as the sum of the two preceding terms.
  • They start with 0 and 1, and the formula is ( F_n = F_{n-1} + F_{n-2} ).
• The nth term of an arithmetic sequence is given by ( a_n = a + (n-1)d ).
• The nth term of a geometric sequence is given by ( a_n = a \cdot r^{(n-1)} ).
• The sum of the first n terms of an arithmetic series is ( S_n = \frac{n}{2} (2a + (n-1)d) ) or ( S_n = \frac{n}{2} (a + l) ), where (l) is the last term.
• The sum of the first n terms of a geometric series is ( S_n = a \frac{1 - r^n}{1 - r} ) (for ( r \neq 1 )).
• Convergence means a sequence or series approaches a limit.
• Divergence means a sequence or series does not approach a limit.
• Infinite Series are sums of infinite sequences, which can converge or diverge.
• Ratio Test, Root Test, Integral Test, and Comparison Test are used to determine convergence of infinite series.
• Applications of sequences and series include calculus, physics, finance, computer science, and statistics.

Test your understanding of sequences and series, including arithmetic, geometric, harmonic, and Fibonacci sequences. Evaluate different types and formulas related to these concepts. Ideal for students looking to strengthen their grasp on this mathematical topic.